Normal distribution.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import quad as integrate\n",
    "from scipy.special import binom\n",
    "from scipy.optimize import fsolve as solve\n",
    "import numpy as np\n",
    "from numpy import sqrt, sin, cos, pi, exp, inf, round, ceil\n",
    "from numpy import array as arr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
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       "     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
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       "\n",
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       "\n",
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       "  }\n",
       "}(this));"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "from bokeh.io import output_notebook, show\n",
    "from bokeh.plotting import figure\n",
    "output_notebook()\n",
    "\n",
    "def fig():\n",
    "    return figure(plot_width=950, plot_height=500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the normal distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def normal_distribution(mu, sigma):\n",
    "    def f(x):\n",
    "        return 1/sqrt(2*pi*sigma**2)*exp(-(x-mu)**2/(2*sigma**2))\n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999984, 1.6699038772828793e-09)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(normal_distribution(12,3), -inf, inf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the binomial distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def binomial_distribution(n):\n",
    "    def f(x):\n",
    "        return binom(n,x)*(1/2)**n\n",
    "    return f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define our specific cases:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 20 # Up until now, only works for even n\n",
    "mu = int(n/2)\n",
    "\n",
    "bin = binomial_distribution(n)\n",
    "\n",
    "sigma = sqrt(sum(bin(x)*(mu-x)**2 for x in range(n+1)))\n",
    "\n",
    "norm = normal_distribution(mu, sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some basic plots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_norm = np.linspace(-1, n+1, 1000)\n",
    "y_norm = norm(x_norm)\n",
    "\n",
    "x_bin = np.linspace(0,n,n+1)\n",
    "y_bin = bin(x_bin)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = fig()\n",
    "p.line(x_norm, y_norm)\n",
    "p.vbar(x_bin, top=y_bin, width=0.8)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding the corresponding intervals\n",
    "The next step is to find out the intervals of the normal distribution whose integrals yield the same values as the binomial distribution for the corresponding $x$. \n",
    "\n",
    "The function ``integ_around`` returns \n",
    "$$ \\int^{\\mu+a}_{\\mu-a} f(x) \\;dx$$\n",
    "as a function of $a$ for the given $\\mu$ where $f(x)$ is the normal distribution. Optionally fixed upper or lower bounds can be passed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def integ_around(val, upper_bound=False, lower_bound=False):\n",
    "    # Create function\n",
    "    def f(a):\n",
    "        # Set upper and lower bounds depending on whether they are specified or not\n",
    "        upper = val+a if not upper_bound else upper_bound\n",
    "        lower = val-a if not lower_bound else lower_bound\n",
    "        # Return value of integral\n",
    "        return integrate(norm, lower, upper)\n",
    "    return f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function ``diff_to_integ`` returns a function of $a$ which gives the difference of the bionmial distribution at the given $\\mu$ and the integral $ \\int^{\\mu+a}_{\\mu-a} f(x) \\;dx$. Here again optionally fixed upper or lower bounds can be passed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def diff_to_integ(val, upper_bound=False, lower_bound=False):\n",
    "    integ = integ_around(val, upper_bound, lower_bound)\n",
    "    def f(a):\n",
    "        return integ(a) - bin(val)\n",
    "    return f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we have to find the zeros of ``diff_to_integ`` for different $\\mu$.\n",
    "\n",
    "The interval endpoints are collected in the array ``intervals``. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "intervals = []\n",
    "\n",
    "# The first\n",
    "a = solve(diff_to_integ(mu), 0)[0]\n",
    "intervals.append(mu+a)\n",
    "\n",
    "# The rest\n",
    "for val in range(mu+1, n):\n",
    "    lower_bound = intervals[-1]\n",
    "    a = solve(diff_to_integ(val, lower_bound=lower_bound), 0)[0]\n",
    "    intervals.append(val+a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10.497874815138442,\n",
       " 11.496107575745944,\n",
       " 12.501972300399022,\n",
       " 13.521207990540706,\n",
       " 14.560700324772521,\n",
       " 15.629378769106374,\n",
       " 16.73989052048239,\n",
       " 17.912263047995339,\n",
       " 19.183902573892187,\n",
       " 20.650394089291296]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "intervals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "other_side = n - arr(intervals)[::-1]\n",
    "intervals = np.concatenate((other_side, intervals))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let us take a look at how it looks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
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GTThKaz+D2FqEuw5sP3QS/XEO2Gw/Qsr9eEmmbT+Df/lLhXluP0m6Fu3aUW8/FAfc1rEXcD8j/RCXHIlwP4Quiq46/XA/x5mrOxl0cT8vLp2Dxe1xPwMLBPJManI/w/i0GL3pcj8c+V+vI2xzPyDKNJOO8XM/szmAxgt6dD8HJ0JwqQV1P/0OvNt1lHU/hwL4d38mdj/L5EfX1Lt2P2rQvK6EVHc/boOW1Z3wdz+ssapEL5B4P5gexBVIM3k/kV/5gvfZeT+mKPvlTIR6P7cGWrdXMns/xWnDjSfkez805DUdzJl8P+GDLDZVU30/qyrBxNIQfj+/zcXPVNJ+P7mC1Hfrl38/HiIre9MwgD+VK0DOy5eAP+UdJOnmAIE/UgunCi1sgT9ov957ptmBP7JPko9bSYI/P86goVS7gj9lFmMWmi+DP8OsCFo0poM/tqzv3ysfhD+SvfchiZqEP0kL0J9UGIU/iT5A3paYhT8BcWxmWBuGPwwcFMWhoIY/uP/LiXsohz8QAjNG7rKHP7QGIo0CQIg/57/W8cDPiD9dexkHMmKJP3rtXV5e94k/U//ehk6Pij8YpLUMCyqLPwG86necx4s/lAuESwtojD9WT4wEYAuNP5F0FhmjsY0/swE899xajj8MuRYEFgePPzKCtZpWto8/yFOGhVM0kD8Su3DMB4+QP6OnWD1M65A/pB5K7CRJkT8ujo3llaiRP5+1FC2jCZI/q2vlvVBskj8fTYKJotCSP2JfUXecNpM/ubIAZEKekz/gEOkgmAeUP7vEbnOhcpQ/U4lgFGLflD8FrlSv3U2VPwx/BOIXvpU/BgKmOxQwlj80FkQ81qOWP/AJFVRhGZc/ArbP4riQlz93Mf824AmYPx4wVY3ahJg/YR/7D6sBmT/EFOLVVICZPxujEeLaAJo/CKv1IkCDmj+lPKtxhwebP6KgTJGzjZs/Wp88LscVnD+9HXHdxJ+cP6QnvRuvK50/X4AaTYi5nT9t0vK7UkmeP/iXaJgQ254/g9af98Nunz8hZINpNwKgPzNHz4MJTqA/GwSiKtmaoD90dfwup+igP52bllJ0N6E/5ZSER0GHoT/v4duvDtihPz0FWR3dKaI/VY0FEa18oj+gmN76ftCiP/DiezlTJaM/X2y3GSp7oz9syVXWA9KjP8Msr5fgKaQ/ojpZc8CCpD+3ttFro9ykP7ccKnCJN6U/tDS0W3KTpT/Ws6/1XfClP1v6+PBLTqY/OgC56zutpj8ogRZvLQ2nP2B56O4fbqc/fwRqyRLQpz9Yr+9GBTOoP7tOnpn2lqg/xmsj3eX7qD+sV28W0mGpPwH4cDO6yKk/zFzTCp0wqj/OMb1beZmqPyocks1NA6s/gBW27xhuqz+Y1VI52dmrP79bHwmNRqw/TKgppTK0rD+Xt6I6yCKtP5TOrN1Lkq0/qCksibsCrj8eHpoeFXSuPwi+2mVW5q4/1A4VDX1Zrz8D4o2ohs2vP8CvQlk4IbA/eqiLRRxcsD9ZHg+8bZewP7OvjFIr07A/LcmKkVMPsT96lkz05EuxP1t3yejdiLE/p/+lzzzGsT/2iC78/wOyPzlcU7QlQrI/TnmmMKyAsj+4Aluckb+yPw9TRhXU/rI/rMHiq3E+sz8bHFRjaH6zPyzZbTG2vrM/bwq7/lj/sz8pEYimTkC0P2ga7vaUgbQ/GmfgsCnDtD9JYzuICgW1P3yQ1SM1R7U/ikaSHaeJtT99TXYCXsy1P+ZTvlJXD7Y/bUP3gZBStj8Sdhj3Bpa2PzLOnwy42bY/K7KvEKEdtz+r7S5Fv2G3P+h46t8Pprc/lCa5CpDqtz8POaHjPC+4Py/e/3wTdLg/75Gy3RC5uD+QZkIBMv64P48xEdhzQ7k/CZyIR9OIuT9HFEsqTc65Pw+fZlDeE7o/loWJf4NZuj+X3ThzOZ+6P5LoCN385Lo/BkbXZMoquz+c9QapnnC7Pzgkvj52trs//78lsk38uz/wzaqGIUK8P817QTfuh7w/EempNrDNvD88oLbvYxO9P9a6lMUFWb0/yKgVFJKevT9elPovBeS9P35bQWdbKb4//hVzAZFuvj+DIfQ/orO+P+6pVV6L+L4/0qWokkg9vz/9PdIN1oG/Px2W4fsvxr8/s3YzQikFwD9NBmblHCfAP5L7VvfwSMA/h+obhqNqwD/whD+eMozAPxc38EqcrcA/p08vlt7OwD/prACJ9+/AP1bpmivlEME/FQGYhaUxwT9DaiaeNlLBP2mZOnyWcsE/j+vAJsOSwT9T78+kurLBP/YF2/160sE/tlXlOQLywT8wBrVhThHCP76/Bn9dMMI/q2bBnC1Pwj+xCirHvG3CP5sDGAwJjMI/XDMpexCqwj8QZvYl0cfCP3HIRyBJ5cI/v2xJgHYCwz+x1r9eVx/DP1eGPNfpO8M/M3pSCCxYwz+yoMoTHHTDP+kw2B64j8M/0eJMUv6qwz/e/sza7MXDPxI9A+mB4MM/bGvUsbv6wz++05JumBTEP+hYMV0WLsQ/a0N2wDNHxD9htS3g7l/EP9m9WwlGeMQ/nQNujjeQxD+a/2zHwafEP8++LBLjvsQ/NiR90pnVxD+iolly5OvEP/FmGGLBAcU/A+uYGC8XxT+v6XETLCzFP2SsHte2QMU/4aor781UxT/HdWLub2jFP83k9G6be8U/TIGnEk+OxT9NJvuCiaDFPwfPVXFJssU/Co0ql43DxT+BnyC2VNTFP8+kOZid5MU/UeD2D2f0xT/fjX34rwPGP+s8ujV3EsY/ZS2DtLsgxj9mqLlqfC7GPyFPale4O8Y/glrsgm5Ixj80xv/+nVTGP/Jh6uZFYMY/E8STX2Vrxj+nGKCX+3XGP3rIiccHgMY/l/O5MYmJxj8nu58if5LGP5BWxvDomsY/J/Dp/MWixj/DRQuyFarGP9MJgoXXsMY/2AEO9wq3xj883+aQr7zGP9nPyufEwcY/mMMLm0rGxj/oZJtUQMrGPwHCFcmlzcY//aTKt3rQxj9OmcXqvtLGPwud1DZy1MY/AX2Ne5TVxj+K2lGjJdbGP4raUaMl1sY/AX2Ne5TVxj8LndQ2ctTGP06Zxeq+0sY//aTKt3rQxj8BwhXJpc3GP+hkm1RAysY/mMMLm0rGxj/Zz8rnxMHGPzzf5pCvvMY/2AEO9wq3xj/TCYKF17DGP8NFC7IVqsY/KPDp/MWixj+QVsbw6JrGPye7nyJ/ksY/l/O5MYmJxj96yInHB4DGP6cYoJf7dcY/E8STX2Vrxj/yYermRWDGPzTG//6dVMY/glrsgm5Ixj8iT2pXuDvGP2aouWp8LsY/ZS2DtLsgxj/rPLo1dxLGP9+NffivA8Y/UuD2D2f0xT/PpDmYneTFP4GfILZU1MU/Co0ql43DxT8Hz1VxSbLFP04m+4KJoMU/TIGnEk+OxT/N5PRum3vFP8d1Yu5vaMU/4aor781UxT9krB7XtkDFP6/pcRMsLMU/A+uYGC8XxT/xZhhiwQHFP6KiWXLk68Q/NiR90pnVxD/PviwS477EP5r/bMfBp8Q/nQNujjeQxD/ZvVsJRnjEP2G1LeDuX8Q/a0N2wDNHxD/oWDFdFi7EP77Tkm6YFMQ/bGvUsbv6wz8SPQPpgeDDP97+zNrsxcM/0eJMUv6qwz/pMNgeuI/DP7KgyhMcdMM/M3pSCCxYwz9XhjzX6TvDP7PWv15XH8M/v2xJgHYCwz9xyEcgSeXCPxBm9iXRx8I/XDMpexCqwj+dAxgMCYzCP7EKKse8bcI/q2bBnC1Pwj++vwZ/XTDCPzAGtWFOEcI/uFXlOQLywT/2Bdv9etLBP1Pvz6S6ssE/j+vAJsOSwT9pmTp8lnLBP0ZqJp42UsE/FQGYhaUxwT9W6Zor5RDBP+msAIn378A/p08vlt7OwD8aN/BKnK3AP/CEP54yjMA/h+obhqNqwD+S+1b38EjAP00GZuUcJ8A/s3YzQikFwD8dluH7L8a/P/090g3Wgb8/0qWokkg9vz/uqVVei/i+P4Mh9D+is74//hVzAZFuvj9+W0FnWym+P16U+i8F5L0/yKgVFJKevT/WupTFBVm9Pzygtu9jE70/EempNrDNvD/Ne0E37oe8P/DNqoYhQrw//78lsk38uz84JL4+dra7P5z1BqmecLs/BkbXZMoquz+S6Ajd/OS6P5fdOHM5n7o/loWJf4NZuj8Wn2ZQ3hO6P0cUSypNzrk/CZyIR9OIuT+PMRHYc0O5P5BmQgEy/rg/9ZGy3RC5uD8v3v98E3S4Pw85oeM8L7g/lCa5CpDqtz/oeOrfD6a3P7HtLkW/Ybc/K7KvEKEdtz8yzp8MuNm2PxJ2GPcGlrY/bUP3gZBStj/sU75SVw+2P31NdgJezLU/ikaSHaeJtT98kNUjNUe1P0ljO4gKBbU/H2fgsCnDtD9oGu72lIG0PykRiKZOQLQ/bwq7/lj/sz8s2W0xtr6zPxscVGNofrM/rMHiq3E+sz8PU0YV1P6yP7gCW5yRv7I/TnmmMKyAsj83XFO0JUKyP/aILvz/A7I/qf+lzzzGsT9bd8no3YixP32WTPTkS7E/K8mKkVMPsT+zr4xSK9OwP10eD7xtl7A/eqiLRRxcsD/Cr0JZOCGwP/3hjaiGza8/1A4VDX1Zrz8MvtplVuauPx4emh4VdK4/ryksibsCrj+RzqzdS5KtP5e3ojrIIq0/UKgppTK0rD+/Wx8JjUasP57VUjnZ2as/fBW27xhuqz8qHJLNTQOrP9IxvVt5mao/zFzTCp0wqj8I+HAzusipP6xXbxbSYak/xmsj3eX7qD+/Tp6Z9paoP1iv70YFM6g/gwRqyRLQpz9geejuH26nPyiBFm8tDac/PwC56zutpj9b+vjwS06mP9mzr/Vd8KU/tDS0W3KTpT+3HCpwiTelP7q20Wuj3KQ/ojpZc8CCpD/HLK+X4CmkP2zJVdYD0qM/X2y3GSp7oz/t4ns5UyWjP6CY3vp+0KI/WY0FEa18oj89BVkd3SmiP+/h268O2KE/45SER0GHoT+dm5ZSdDehP3d1/C6n6KA/GwSiKtmaoD82R8+DCU6gPx5kg2k3AqA/g9af98Nunz/9l2iYENueP23S8rtSSZ4/ZYAaTYi5nT+gJ70bryudP70dcd3En5w/Yp88LscVnD+ioEyRs42bP6w8q3GHB5s/Aqv1IkCDmj8boxHi2gCaP8gU4tVUgJk/YR/7D6sBmT8kMFWN2oSYP3Qx/zbgCZg/ArbP4riQlz/zCRVUYRmXPzQWRDzWo5Y/CQKmOxQwlj8MfwTiF76VPwWuVK/dTZU/W4lgFGLflD+7xG5zoXKUP+IQ6SCYB5Q/ubIAZEKekz9iX1F3nDaTPyRNgomi0JI/q2vlvVBskj+jtRQtowmSPy6OjeWVqJE/pB5K7CRJkT+np1g9TOuQPxK7cMwHj5A/zlOGhVM0kD8ygrWaVraPPwy5FgQWB48/sAE899xajj+RdBYZo7GNP2JPjARgC40/oguESwtojD/wu+p3nMeLPw+ktQwLKos/U//ehk6Pij+B7V1eXveJP2h7GQcyYok/3b/W8cDPiD+rBiKNAkCIPxACM0busoc/u//LiXsohz8YHBTFoaCGP/lwbGZYG4Y/gz5A3paYhT9JC9CfVBiFP5S99yGJmoQ/wKzv3ysfhD++rAhaNKaDP2EWYxaaL4M/P86goVS7gj+1T5KPW0mCP3K/3num2YE/TgunCi1sgT/fHSTp5gCBP5UrQM7Ll4A/IyIre9MwgD/NgtR365d/P7bNxc9U0n4/qyrBxNIQfj/hgyw2VVN9PzvkNR3MmXw/1mnDjSfkez+wBlq3VzJ7P6Yo++VMhHo/kV/5gvfZeT+cHsQVSDN5P7yxqkQvkHg/ZoOW1Z3wdz9q0LyuhFR3P8vkR9fUu3Y/jAL4d38mdj8KD7zbdZR1P/8mQnCpBXU/szmAxgt6dD8gyjSTjvFzPyX5X68jbHM/zvi0GL3pcj/8CgTyTGpyPy8unYPF7XE/x5mrOxl0cT+MLoquOv1wPy39EJcciXA/Cwfc1rEXcD9Juhbt2lFvP4t/+UuFeW4/Scr9eEmmbT+GEv1xDthsP3nYWoS7Dmw/AMdGTThKaz+MQvO5bIpqPymwwAdBz2k/GrtdxJ0YaT+O4tzNa2ZoPyKYv1KUuGc/vjX30QAPZz9zFdwam2lmP18QG01NyGU/Ba6Z2AErZT/YSFF9o5FkP4ZwIUsd/GM/vc6ZoVpqYz/707svR9xiPwVytPPOUWI/8CaOOt7KYT/5ndufYUdhPxwoWw1Gx2A//E6TunhKYD9UiNFYzqFfPzPYW2n+tF4/EI5Y413OXT8R6k+bye1cPz0pu/weE1w/H3P1CDw+Wz9abiJW/25aP733Cg5IpVk/Em7w7PXgWD8KCFdA6SFYP++lx+UCaFc/9o2JSSSzVj+yglRlLwNWP7We+74GWFU/wV4RZ42xVD8uRIX3pg9UP7xyO5I3clM/IbCe3yPZUj+RJS0NUURSP/FCAMyks1E/BCJQTwUnUT9Cw/JKWZ5QP7t/1/GHGVA/IQv+6PEwTz8raeD+KDZOPwldVHOGQk0/F9QnhtxVTD+/MtNl/m9LP+ReUCzAkEo/sJ3u2/a3ST/p1SNceOVIP9XBW3YbGUg/dpvF0rdSRz8PxiD1JZJGP3j1iDk/10U/hlBC0d0hRT8GBYa/3HFEP8PDT9YXx0M/0pMss2shQz+saQu8tYBCP9DrDxzU5EE/98dnwKVNQT9t/CJVCrtAPyZxD0LiLEA/YnYwTx1GPz+IwlG34jo+P3vkKc3ZNz0/wn4IAMk8PD9KCMgXeEk7P06Jui6wXTo/dlChqzt5OT+oIbA75ps4P5penMx8xTc/C564hs31Nj8mIx3Hpyw2P52g3RncaTU/HapMNDytND9RNk3vmvYzP1KIskHMRTM/t9WuOqWaMj+891D8+/QxP5tvEbanVDE/sANvn4C5MD+kM5vyXyMwP/V6bc4/JC8/+NM8WjcLLj+qnpDOXvssP6BwY01w9Cs/r6y7yif2Kj+gxMoCQwAqP0qYLnCBEik/RSZVQqQsKD8dwAJUbk4nPz/n+iGkdyY/cvPLwQuoJT8MnL3YbN8kP6104pKQHSQ/5WpMmkFiIz9fSGQOTK0iP9w4ZHt9/iE/0E310aRVIT/V8O9ekrIgP3U1P8MXFSA/6ODN1w/6Hj/C81gUbtQdP9Hswxf1uBw/FAioJ1KnGz80PLPkNJ8aP9QUDDxPoBk/hzn5WFWqGD8qMsyW/bwXP9D2DnMA2BY/jMzzfxj7FT8q7gZXAiYVP+d3IYx8WBQ/3wWdoEeSEz/Ocsf2JdMSP2IclsXbGhI/Aw6YDC9pET/qbyaI570QP9OQ0qXOGBA/MsEh8l7zDj9GVUBgrcANPwG7WBIlmQw/7lJYj2V8Cz90d/FTEWoKPzu3Pb7NYQk/klTR+UJjCD8ziD7sG24HP0UDByIGggY/EyX6u7GeBT8XW/5c0cMEP2UaRBga8QM/++HgX0MmAz+ps9DzBmMCP8dsXNEgpwE/QWzjIk/yAD+/7AYwUkQAP1v7apzYOf8+jgcqo8P3/T7mUpP88cH8PoRUMPTzl/s+KsbDfF15+j7iJPIUxmX5Pqttl6zIXPg+6+/GigNe9z5pGnI0GGn2Pnsks1OrffU+OH64n2Sb9D689U3F7sHzPo6EAFD38PI+pb/Zky4o8j4B6a+XR2fxPmSoBwD4rfA+vPEI9e/37z5SuslTBKLuPrXvDiWnWe0+11DsDlce7D5PsOgvl+/qPohSG/vuzOk+tYREFeq16D4i9NsyGKrnPhBcD/cMqeY+ZzOs01+y5T43Fe/pq8XkPnavM+yP4uM+iBmAAK4I4z4lguajqzfiPmQ6uI4xb+E+OzWFmeuu4D4WScZFEe3fPs3n7ut0i94+EOB+Y2s43T4tACRiY/PbPkPSrObQu9o+MErXCy2R2T5r3Xfc9XLYPmdl8CiuYNc+BVnuXd1Z1j58EGhcD17VPr7q0FLUbNQ+6lZ8l8CF0z5V5ieEbKjSPqm0pFJ01NE+1pmY+ncJ0T7HvE8QG0fQPiKRLkkJGs8+U0s0S761zT7zg3GBsGDMPlUsqopBGss+DnIYEdnhyT7P9a2T5LbIPiJDFjHXmMc+lwdudCmHxj5izaIjWYHFPhc+bw/phsQ+qT7o5GCXwz5Xco8ATbLCPm3530I+18E+0X9K5skFwT5T9JVWiT3APpL7NBM0/L4+6AiZsjqOvT4kyxu+cDC8PsZAd6An4ro+4xzdt7eiuT4P3csTgHG4PgiQJTXmTbc+58l20FU3tj64sFySQC21PgCC+eUdL7Q+h1hnvWo8sz7gaBlcqVSyPn1UHCNhd7E+25YmXx6ksD4=\",\"dtype\":\"float64\",\"shape\":[1000]}}},\"id\":\"2d8e5c4d-b354-415c-8b73-88d6f604bc3c\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"below\":[{\"id\":\"6e1232cb-9986-4a29-86cd-494c9bdf7242\",\"type\":\"LinearAxis\"}],\"left\":[{\"id\":\"c0c84609-3e1b-4771-8f73-fb29786f51ad\",\"type\":\"LinearAxis\"}],\"plot_height\":500,\"plot_width\":950,\"renderers\":[{\"id\":\"6e1232cb-9986-4a29-86cd-494c9bdf7242\",\"type\":\"LinearAxis\"},{\"id\":\"1fa2242e-92c8-4050-a3d3-338c82aeab8b\",\"type\":\"Grid\"},{\"id\":\"c0c84609-3e1b-4771-8f73-fb29786f51ad\",\"type\":\"LinearAxis\"},{\"id\":\"8df99790-feb1-4f50-a4fd-e423d26acdc7\",\"type\":\"Grid\"},{\"id\":\"68e641d9-13a7-4c5b-ad82-a9cd8dfd9976\",\"type\":\"BoxAnnotation\"},{\"id\":\"d0238d90-c615-469b-b702-2a341cdb1e20\",\"type\":\"GlyphRenderer\"},{\"id\":\"d8e76811-bfb8-4353-8c45-7664ef9a893b\",\"type\":\"GlyphRenderer\"}],\"title\":{\"id\":\"36d5bbd8-144a-4ae4-84de-6c033716679a\",\"type\":\"Title\"},\"tool_events\":{\"id\":\"5e085bdc-e821-4cbe-b11f-4ddfee9b0fd7\",\"type\":\"ToolEvents\"},\"toolbar\":{\"id\":\"e3b8145c-8d9c-4529-91bf-43a9a040412f\",\"type\":\"Toolbar\"},\"x_range\":{\"id\":\"97f8ba80-ee96-46ce-9c56-318a9da49a3b\",\"type\":\"DataRange1d\"},\"y_range\":{\"id\":\"5078f40b-9cd6-45d2-b2ce-91d0e3add88c\",\"type\":\"DataRange1d\"}},\"id\":\"171d3f2c-a3d9-4f0b-9c30-c6e46a48261d\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{},\"id\":\"5e085bdc-e821-4cbe-b11f-4ddfee9b0fd7\",\"type\":\"ToolEvents\"}],\"root_ids\":[\"171d3f2c-a3d9-4f0b-9c30-c6e46a48261d\"]},\"title\":\"Bokeh 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       "            var render_items = [{\"docid\":\"7801f542-826b-45f4-99cf-9357d9b56681\",\"elementid\":\"a249fc9d-412d-4eef-b9ea-9749a1b96e4c\",\"modelid\":\"171d3f2c-a3d9-4f0b-9c30-c6e46a48261d\"}];\n",
       "            \n",
       "            Bokeh.embed.embed_items(docs_json, render_items);\n",
       "          };\n",
       "          if (document.readyState != \"loading\") fn();\n",
       "          else document.addEventListener(\"DOMContentLoaded\", fn);\n",
       "        })();\n",
       "      },\n",
       "      function(Bokeh) {\n",
       "      }\n",
       "    ];\n",
       "  \n",
       "    function run_inline_js() {\n",
       "      \n",
       "      if ((window.Bokeh !== undefined) || (force === true)) {\n",
       "        for (var i = 0; i < inline_js.length; i++) {\n",
       "          inline_js[i](window.Bokeh);\n",
       "        }if (force === true) {\n",
       "          display_loaded();\n",
       "        }} else if (Date.now() < window._bokeh_timeout) {\n",
       "        setTimeout(run_inline_js, 100);\n",
       "      } else if (!window._bokeh_failed_load) {\n",
       "        console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
       "        window._bokeh_failed_load = true;\n",
       "      } else if (force !== true) {\n",
       "        var cell = $(document.getElementById(\"a249fc9d-412d-4eef-b9ea-9749a1b96e4c\")).parents('.cell').data().cell;\n",
       "        cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "      }\n",
       "  \n",
       "    }\n",
       "  \n",
       "    if (window._bokeh_is_loading === 0) {\n",
       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "      run_inline_js();\n",
       "    } else {\n",
       "      load_libs(js_urls, function() {\n",
       "        console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "        run_inline_js();\n",
       "      });\n",
       "    }\n",
       "  }(this));\n",
       "</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = fig()\n",
    "p.line(x_norm, y_norm)\n",
    "p.vbar(intervals, top=norm(intervals), width=0.02)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Seems to be quite sensible. Now, let us change the standard deviation of the normal distribution, but compute the integrals with the same upper and lower bounds respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "# \"nnorm\" for \"narrower norm\" or \"new norm\" as one prefers\n",
    "nnorm = normal_distribution(n/2, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
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       "       \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
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\",\"dtype\":\"float64\",\"shape\":[1000]},\"y\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"shape\":[1000]}}},\"id\":\"5c78977c-6e9c-44b7-927e-99ed62b7d8f0\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"line_alpha\":{\"value\":0.1},\"line_color\":{\"value\":\"#1f77b4\"},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"f87dee7e-15f9-40b6-95bf-d62bc148d9c2\",\"type\":\"Line\"},{\"attributes\":{\"formatter\":{\"id\":\"eff38cb4-ca70-4e5a-922a-c2ceb68f15dc\",\"type\":\"BasicTickFormatter\"},\"plot\":{\"id\":\"2ad58f3a-9782-42a1-9794-02bdeb91ae57\",\"subtype\":\"Figure\",\"type\":\"Plot\"},\"ticker\":{\"id\":\"87523718-319a-4e45-a96f-4f34119801a1\",\"type\":\"BasicTicker\"}},\"id\":\"08530cc8-6c87-466e-9fc1-6e31aedacaac\",\"type\":\"LinearAxis\"},{\"attributes\":{\"callback\":null},\"id\":\"4c9db2ee-7dad-4a6c-9f93-ce09a55a75c2\",\"type\":\"DataRange1d\"},{\"attributes\":{\"data_source\":{\"id\":\"d98b3901-9113-43c0-bf65-3682a6b9afb9\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"a49a322e-3d9e-4e24-a7a6-69659e12fe5c\",\"type\":\"VBar\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"cba688db-b74b-43be-9b4d-14b923bbe299\",\"type\":\"VBar\"},\"selection_glyph\":null},\"id\":\"f66c2886-8940-4534-baa0-7e61d73ba14a\",\"type\":\"GlyphRenderer\"},{\"attributes\":{},\"id\":\"460c4308-4118-42a4-ae4b-36a6f5e31cf2\",\"type\":\"ToolEvents\"},{\"attributes\":{\"formatter\":{\"id\":\"1a6b4c71-7736-4de7-9462-6429efb6212e\",\"type\":\"BasicTickFormatter\"},\"plot\":{\"id\":\"2ad58f3a-9782-42a1-9794-02bdeb91ae57\",\"subtype\":\"Figure\",\"type\":\"Plot\"},\"ticker\":{\"id\":\"a0fb004a-25ec-4aa3-82dd-85022097e7e9\",\"type\":\"BasicTicker\"}},\"id\":\"188d6bab-2eff-4ef0-84f4-12f4a26b15e5\",\"type\":\"LinearAxis\"}],\"root_ids\":[\"2ad58f3a-9782-42a1-9794-02bdeb91ae57\"]},\"title\":\"Bokeh 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       "            var render_items = [{\"docid\":\"c36bc9ff-7abc-4136-943f-d935bb9b8018\",\"elementid\":\"b02146d4-02e4-43a4-8632-744eb0d031d2\",\"modelid\":\"2ad58f3a-9782-42a1-9794-02bdeb91ae57\"}];\n",
       "            \n",
       "            Bokeh.embed.embed_items(docs_json, render_items);\n",
       "          };\n",
       "          if (document.readyState != \"loading\") fn();\n",
       "          else document.addEventListener(\"DOMContentLoaded\", fn);\n",
       "        })();\n",
       "      },\n",
       "      function(Bokeh) {\n",
       "      }\n",
       "    ];\n",
       "  \n",
       "    function run_inline_js() {\n",
       "      \n",
       "      if ((window.Bokeh !== undefined) || (force === true)) {\n",
       "        for (var i = 0; i < inline_js.length; i++) {\n",
       "          inline_js[i](window.Bokeh);\n",
       "        }if (force === true) {\n",
       "          display_loaded();\n",
       "        }} else if (Date.now() < window._bokeh_timeout) {\n",
       "        setTimeout(run_inline_js, 100);\n",
       "      } else if (!window._bokeh_failed_load) {\n",
       "        console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
       "        window._bokeh_failed_load = true;\n",
       "      } else if (force !== true) {\n",
       "        var cell = $(document.getElementById(\"b02146d4-02e4-43a4-8632-744eb0d031d2\")).parents('.cell').data().cell;\n",
       "        cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "      }\n",
       "  \n",
       "    }\n",
       "  \n",
       "    if (window._bokeh_is_loading === 0) {\n",
       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "      run_inline_js();\n",
       "    } else {\n",
       "      load_libs(js_urls, function() {\n",
       "        console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "        run_inline_js();\n",
       "      });\n",
       "    }\n",
       "  }(this));\n",
       "</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_nnorm = np.linspace(-1, n+1, 1000)\n",
    "y_nnorm = nnorm(x_nnorm)\n",
    "\n",
    "p = fig()\n",
    "p.line(x_nnorm, y_nnorm)\n",
    "p.vbar(intervals, top=nnorm(intervals), width=0.02)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us add $\\pm\\infty$ to the list of intervals in order for it to be comprehensive:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "intervals = list(intervals)\n",
    "intervals.insert(0, -inf)\n",
    "intervals.append(inf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "vals = []\n",
    "for i in range(len(intervals)-1):\n",
    "    vals.append(integrate(nnorm, intervals[i], intervals[i+1])[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
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\",\"dtype\":\"float64\",\"shape\":[1000]},\"y\":{\"__ndarray__\":\"ytqffrz7JD8IsNu3Io4lPy60fCo4JCY/ETyLSxK+Jj/Iy9j8xlsnP4G3uo5s/Sc/+gvJwRmjKD9JtaHI5UwpP2Tcr0no+ik/b3L3YDmtKj8U3+Sh8WMrP7fIIBkqHyw/oOpnTvzeLD8V7WZGgqMtP000moTWbC4/xJYxDRQ7Lz/I93szKwcwP8Q9n85cczA/yPnlIC3iMD+pqWe3qlMxPygda2TkxzE/1GtjQOk+Mj+9pe6qyLgyPw021kuSNTM/Ku0QFFa1Mz+Ep8Y+JDg0P6qGVVINvjQ/W7BYISJHNT9XiLDLc9M1P3VYjL8TYzY/x1l1uhP2Nj8nElvKhYw3P6v3oE58Jjg/s0wt+QnEOD/aJHnPQWU5PySEoSs3Cjo/Lol5vf2yOj+hkJ2LqV87PzVDh/ROEDw/Z3uirwLFPD/C8WLO2X09PxeeWr3pOj4/OrlQRUj8Pj8iTlmMC8I/PzCidgslRkA/bWiBZI2tQD/wG5NzShdBPwxlyQtog0E/lgOmMfLxQT+mZ5sb9WJCP+ZFmTJ91kI/JhyZEpdMQz80mSqLT8VDP5bc/5+zQEQ/KYB5idC+RD+dXzK1sz9FPz0Qi8Zqw0U/qvo0lwNKRj8dCr03jNNGP+zgFfASYEc/N4YhQKbvRz+hezrgVIJIP2UtvMEtGEk/SK2KD0CxST+fp5kum01KPyGEcr5O7Uo/j6C5mWqQSz8jlbLW/jZMPw5zw8cb4Uw/KOn2+9GOTT+fQH0/MkBOP6YcLJxN9U4/Ru38WTWuTz8SAcV/fTVQP/gURSnYlVA/fGYlrDP4UD/rNhQrmVxRP9u+AukRw1E/RbhfSacrUj/s/VDQYpZSP4M27CJOA1M/sYBuB3NyUz+gFXNl2+NTP7LXKEaRV1Q/1cGG1J7NVD+rLX9dDkZVP3PkMVDqwFU/n+8cPj0+Vj/FHkzbEb5WP/w1h/5yQFc/V7p+oWvFVz/yT/fgBk1YPwqe8/xP11g/gK3cWFJkWT/xtKh7GfRZPxhHABCxhlo/edZh5CQcWz80gkPrgLRbP6kfMzvRT1w/A3T0DiLuXD8VkZ3Ff49dP+xIseL2M14/Y6o3DpTbXj9ae9QUZIZfP6PQ7fM5GmA/h7axTuhyYD/OXSq4Q81gP/RuPOBSKWE/Q55CiRyHYT9OUxKIp+ZhP/Ag/8P6R2I/swXdNh2rYj9obgHtFRBjP1f0QgXsdmM/es/3sKbfYz+w9vIzTUpkP0Lmf+TmtmQ/PAZdK3slZT/pqbSDEZZlP9GgFHuxCGY/F1NksWJ9Zj+pYdnYLPRmP5TD6rUXbWc/3FlCHyvoZz8d8qz9bmVoPySyCEzr5Gg/juQxF6hmaT8rIO59reppP1fD1bADcWo/zrs78rL5aj8QlROWw4RrP/fF1QE+Emw/fzZirCqibD88+OAdkjRtPyUpoe98yW0/sfv1y/Ngbj+Y3BFu//puPwmw36Gol28/IRDtIXwbcD89e3Cge2xwPxw8BkvXvnA/WGE/qJMScT/7jplGtWdxP/CgZbxAvnE/TUOtpzoWcj+KfReup29yP3YtzHyMynI/Y29WyO0mcz9+8IVM0IRzP3UoT8w45HM/AXiqESxFdD8hKXLtrqd0PxhPPzfGC3U/64JFzXZxdT9gei2Uxdh1P9x27na3QXY/rYmmZlGsdj+8q3FamBh3Px6mP0+Rhnc/ccmoR0H2dz9GcsFLrWd4P9lX7Gja2ng/R6Srsc1PeT8X03A9jMZ5P4pVaygbP3o/dvpVk3+5ej/GF0OjvjV7Pyh1Z4Hds3s/W/XjWuEzfD+q/Y1gz7V8PyKatsasOX0/Bl3wxH6/fT9w+dOVSkd+PzSXw3YV0X4//9+sp+Rcfz/PxMlqvep/P2z9L4JSPYA/MJg/XVCGgD/3+17qWtCAP2a1+s10G4E/JsH9rKBngT+nE64s4bSBP+F8iPI4A4I/NOcbpKpSgj998uPmOKOCP0/rImDm9II/dB67tLVHgz+kiQeJqZuDP3rps4DE8IM/XSWUPglHhD81Gntkep6EPzXEEJMa94Q/8MinaexQhT8zYhKG8quFP9aqdoQvCIY/Y04i/6Vlhj+qnF2OWMSGP4YCPshJJIc/ueh3QHyFhz+x+i+I8ueHPzPWyy2vS4g/iSXCvLSwiD+xJWq9BReJP2SayrSkfok/TzFoJJTniT+iVhOK1lGKP698tV9uvYo/jtgdG14qiz/nlc0tqJiLP+GDwwRPCIw/kz1HCFV5jD9F0LObvOuMP7HhQR2IX40/KVnR5bnUjT+ZjbJIVEuOP5X7bpNZw44/EIaRDcw8jz84RG74rbePPwPwdMcAGpA/0UOgguRYkD/gOmZEg5iQP4YJ5h/e2JA/edqTJPYZkT8VfhtezFuRP5frQtRhnpE/3pXMirfhkT+clVmBziWSP7KqS7OnapI/vxanF0Swkj/HUvSgpPaSP7ihIT3KPZM/P4Jk1bWFkz+QARtOaM6TP3jxrIbiF5Q/4gNtWSVilD/GzXmbMa2UPwy0nhwI+ZQ/acQ0p6lFlT+ffAMAF5OVP5WCIeZQ4ZU/Tk/VElgwlj+wz3U5LYCWPx39SgfR0JY/rXBuI0Qilz9686suh3SXP2UOYsOax5c/TpxidX8bmD8bYdPRNXCYP1uoDl++xZg/L+6DnBkcmT9clZgCSHOZPw6tiAJKy5k/aclHBiAkmj+O8WFwyn2aPwOm3JtJ2Jo/eAIY3J0zmz+r/a98x4+bP6LKXcHG7Js/CV7Z5ZtKnD+1GrsdR6mcP5yoXZTICJ0/5vi/bCBpnT+IemfBTsqdP2mCQqRTLJ4/H+qKHi+Pnj+b6Kgw4fKeP58nFtJpV58/kRlB8ci8nz9QSbg5fxGgPzrUUxoFRaA/4O3EA/Z4oD+Ax5/aUa2gP6tmrX4Y4qA/mjLeykkXoT/urDyV5UyhP4BX4K7rgqE//Mjg41u5oT/I8Uj7NfChP/2SCrd5J6I/Gunx0yZfoj/6i5kJPZeiP9eFXgq8z6I/26JUg6MIoz8R+zoc80GjPzK4cHeqe6M/ExnqMcm1oz9AtCXjTvCjP277IR07K6Q/bQFTbI1mpD81hJhXRaKkP5w8NGBi3qQ/dXXAAeQapT956yayyVelP9P3l+ESlaU/rgaC+r7SpT9xW4lhzRCmPzAkgHU9T6Y/491ejw6Opj/hCj0CQM2mPx89ShvRDKc/1HXHIcFMpz/K2wBXD42nPxXKR/a6zac/SzjtNMMOqD8UfjxCJ1CoPyBzdkfmkag/LO3MZ//TqD9HnV7AcRapP7tNM2g8Wak/EII4cF6cqT9Aej7j1t+pP5Ka9cWkI6o/TTnsFsdnqj9p04zOPKyqP5qpHN8E8ao/xsa6NB42qz8ecV+1h3urP+AG3EBAwas/+EfbsEYHrD9xDeLYmU2sP9lvUIY4lKw/jF1jgCHbrD/HoTaIUyKtP7Bdx1jNaa0/1/P2po2xrT9yZ44hk/mtP/AvQnHcQa4/oIG2OGiKrj92DIQUNdOuPzExPZtBHK8/Fq5zXYxlrz9zw77lE6+vP8HQwbjW+K8/ebWZqmkhsD8gdvIZhEawPx0/5eO5a7A/Js8BPwqRsD+cDvJedLawP/jMfnT327A/9buUrZIBsT+uqEk1RSexP9Dy4TMOTbE/20HWzuxysT+geNko4JixP/vm3mHnvrE/u7kglwHlsT/gqCbjLQuyP/njzF1rMbI/vDxLHLlXsj+pjzwxFn6yP59qpqyBpLI/Y/EAnPrKsj/A/z4KgPGyP0KJ1v8QGLM/OjbJgqw+sz/aPa2WUWWzPzV9tjz/i7M/xMq/c7Sysz9HhlQ4cNmzP4tkuoQxALQ/wnb7UPcmtD8nbfCSwE20P3YUSz6MdLQ/sw2hRFmbtD8CwHaVJsK0P+aDSh7z6LQ/bwegyr0PtT/56guEhTa1P7qVPzJJXbU/i0EVuweEtT+hPZwCwKq1PxhnJetw0bU/JNdPVRn4tT/lxRUguB62Pzeh2ShMRbY/FFdzS9Rrtj8z0j1iT5K2P8aoJEa8uLY/+Puxzhnftj+whxzSZgW3P7rhVSWiK7c/NecYnMpRtz/FV/gI33e3P0ydbT3enbc/SL/nCcfDtz8mgdo9mOm3PxCqzadQD7g/uXVsFe80uD+/LJVTclq4P7TjaC7Zf7g/4l9bcSKluD9lIEPnTMq4P9iKaVpX77g/IzqblEAUuT9objhfBzm5PwKdRYOqXbk/Dx98ySiCuT/A/Vr6gKa5P8DaN96xyrk/3fNPPbruuT9mQNnfmBK6PwunE45MNro/IUtaENRZuj+Y7zQvLn26P8duabNZoLo/VEUNZlXDuj8GL5cQIOa6P0DU8Hy4CLs/c4aIdR0ruz+JCmPFTU27P31vLThIb7s/9e9OmguRuz9t3fq4lrK7P0uTQmLo07s/wnAnZf/0uz+216yR2hW8P1Mv6rh4Nrw//ugcrdhWvD/JhbpB+Xa8P1WbgkvZlrw/T9aQoHe2vD8x+W4Y09W8P8zVJozq9Lw/5z9U1rwTvT+09zbTSDK9P1SKxGCNUL0/CCe6XoluvT+QZ66uO4y9PwYKIzSjqb0/7pqW1L7GvT+tDZZ3jeO9Pw9CzgYOAL4/YXUdbj8cvj9XnaSbIDi+P4ir2H+wU74/xbeTDe5uvj/SDyY62Im+PyErZ/1tpL4/zYDGUa6+vj+ePlw0mNi+P2Lf+aQq8r4/Lp86pmQLvz8mzJM9RSS/PyfyZHPLPL8/DuAHU/ZUvz/2hODqxGy/PzykbEw2hL8/k15TjEmbvz/6jnTC/bG/PxP69wlSyL8/ck5cgUXevz+W9IVK1/O/P5LWZkWDBMA/AH6HNekOwD+NL9sLHRnAP7CnZ2AeI8A/YgUJzewswD/5knjthzbAPy5yU1/vP8A/1iohwiJJwD98G1q3IVLAP23LbeLrWsA/lR3J6IBjwD97Y9xx4GvAPwJQIScKdMA/HskgtP17wD8tmHjGuoPAP0X44A1Bi8A/9QEyPJCSwD8L9GgFqJnAP8dYrR+IoMA/CAdWQzCnwD8A/+0qoK3AP+chOZPXs8A/UcQ4O9a5wD+jGjDkm7/AP0V/qFEoxcA/MJJ1SXvKwD9dMbmTlM/AP85J5/pz1MA/vYDJSxnZwD+ttIJVhN3AP+1Vkum04cA/VpbX26rlwD/nb5QCZunAP+2BcDbm7MA/icR7UivwwD83EjE0NfPAPyuHeLsD9sA/PLapypb4wD80s41G7vrAP0XyYBYK/cA/e/zUI+r+wD8L+RFbjgDBPz4LuKr2AcE/+YTgAyMDwT+l7R5aEwTBP3jdgaPHBME/8ayT2D8FwT+J+Fr0ewXBP4n4WvR7BcE/8ayT2D8FwT943YGjxwTBP6XtHloTBME/+YTgAyMDwT8+C7iq9gHBPwv5EVuOAME/e/zUI+r+wD9F8mAWCv3APzSzjUbu+sA/PLapypb4wD8rh3i7A/bAPzcSMTQ188A/icR7UivwwD/tgXA25uzAP+dvlAJm6cA/VpbX26rlwD/tVZLptOHAP620glWE3cA/vYDJSxnZwD/OSef6c9TAP10xuZOUz8A/MJJ1SXvKwD9Ff6hRKMXAP6MaMOSbv8A/UcQ4O9a5wD/nITmT17PAPwD/7SqgrcA/CQdWQzCnwD/HWK0fiKDAPwv0aAWomcA/9QEyPJCSwD9F+OANQYvAPy6YeMa6g8A/HskgtP17wD8CUCEnCnTAP3tj3HHga8A/lR3J6IBjwD9ty23i61rAP3wbWrchUsA/1iohwiJJwD8uclNf7z/AP/mSeO2HNsA/YgUJzewswD+wp2dgHiPAP40v2wsdGcA/AH6HNekOwD+S1mZFgwTAP5b0hUrX878/ck5cgUXevz8T+vcJUsi/P/qOdML9sb8/k15TjEmbvz88pGxMNoS/P/aE4OrEbL8/DuAHU/ZUvz8n8mRzyzy/PybMkz1FJL8/Lp86pmQLvz9i3/mkKvK+P6A+XDSY2L4/zYDGUa6+vj8hK2f9baS+P9IPJjrYib4/xbeTDe5uvj+Kq9h/sFO+P1edpJsgOL4/YXUdbj8cvj8PQs4GDgC+P60NlneN470/8JqW1L7GvT8GCiM0o6m9P5Bnrq47jL0/CCe6XoluvT9UisRgjVC9P7b3NtNIMr0/5z9U1rwTvT/M1SaM6vS8PzH5bhjT1bw/T9aQoHe2vD9Xm4JL2Za8P8mFukH5drw//ugcrdhWvD9TL+q4eDa8P7bXrJHaFbw/wnAnZf/0uz9Lk0Ji6NO7P23d+riWsrs/9e9OmguRuz99by04SG+7P4kKY8VNTbs/c4aIdR0ruz9A1PB8uAi7PwYvlxAg5ro/VEUNZlXDuj/HbmmzWaC6P5jvNC8ufbo/IUtaENRZuj8LpxOOTDa6P2ZA2d+YEro/3fNPPbruuT/A2jfescq5P8D9WvqAprk/Dx98ySiCuT8CnUWDql25P2huOF8HObk/IzqblEAUuT/bimlaV++4P2UgQ+dMyrg/4l9bcSKluD+042gu2X+4P78slVNyWrg/vHVsFe80uD8Qqs2nUA+4PyaB2j2Y6bc/SL/nCcfDtz9MnW093p23P8hX+Ajfd7c/NecYnMpRtz+64VUloiu3P7CHHNJmBbc/+Puxzhnftj/JqCRGvLi2PzPSPWJPkrY/FFdzS9Rrtj83odkoTEW2P+XFFSC4HrY/KddPVRn4tT8YZyXrcNG1P6E9nALAqrU/i0EVuweEtT+6lT8ySV21P/nqC4SFNrU/bwegyr0PtT/mg0oe8+i0PwLAdpUmwrQ/sw2hRFmbtD90FEs+jHS0Pydt8JLATbQ/xHb7UPcmtD+LZLqEMQC0P0qGVDhw2bM/w8q/c7Sysz81fbY8/4uzP9w9rZZRZbM/OjbJgqw+sz9Eidb/EBizP7//PgqA8bI/Y/EAnPrKsj+haqasgaSyP6mPPDEWfrI/vjxLHLlXsj/448xdazGyP+CoJuMtC7I/vLkglwHlsT/75t5h576xP6J42SjgmLE/2kHWzuxysT/Q8uEzDk2xP6+oSTVFJ7E/9buUrZIBsT/6zH5099uwP5wO8l50trA/Js8BPwqRsD8fP+XjuWuwPyB28hmERrA/e7WZqmkhsD/B0MG41vivP3PDvuUTr68/Ga5zXYxlrz8xMT2bQRyvP3gMhBQ1064/oIG2OGiKrj/wL0Jx3EGuP3dnjiGT+a0/1/P2po2xrT+0XcdYzWmtP8ehNohTIq0/jF1jgCHbrD/Wb1CGOJSsP3EN4tiZTaw/+0fbsEYHrD/gBtxAQMGrPx5xX7WHe6s/wsa6NB42qz+aqRzfBPGqP2vTjM48rKo/TTnsFsdnqj+VmvXFpCOqPz16PuPW36k/EII4cF6cqT++TTNoPFmpP0edXsBxFqk/Lu3MZ//TqD8dc3ZH5pGoPxR+PEInUKg/UDjtNMMOqD8Vykf2us2nP83bAFcPjac/0XXHIcFMpz8fPUob0QynP+MKPQJAzaY/491ejw6Opj8zJIB1PU+mP29biWHNEKY/rgaC+r7SpT/W95fhEpWlP3nrJrLJV6U/eHXAAeQapT+cPDRgYt6kPzWEmFdFoqQ/cQFTbI1mpD9u+yEdOyukP0O0JeNO8KM/ExnqMcm1oz8yuHB3qnujPxX7OhzzQaM/26JUg6MIoz/ahV4KvM+iP/qLmQk9l6I/Gunx0yZfoj//kgq3eSeiP8jxSPs18KE//8jg41u5oT+AV+Cu64KhP+6sPJXlTKE/lzLeykkXoT+rZq1+GOKgP4PHn9pRraA/5O3EA/Z4oD811FMaBUWgP01JuDl/EaA/kRlB8ci8nz+hJxbSaVefP6LoqDDh8p4/GeqKHi+Pnj9jgkKkUyyeP4h6Z8FOyp0/6fi/bCBpnT+lqF2UyAidP68aux1HqZw/BF7Z5ZtKnD+iyl3BxuybP6/9r3zHj5s/fgIY3J0zmz//pdybSdiaP4vxYXDKfZo/aclHBiAkmj8RrYgCSsuZP2KVmAJIc5k/Ku6DnBkcmT9WqA5fvsWYPxth09E1cJg/UJxidX8bmD9sDmLDmseXP3bzqy6HdJc/rXBuI0Qilz8d/UoH0dCWP7HPdTktgJY/V0/VElgwlj+RgiHmUOGVP598AwAXk5U/acQ0p6lFlT8OtJ4cCPmUP8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       "    }\n",
       "  \n",
       "    if (window._bokeh_is_loading === 0) {\n",
       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
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       "      load_libs(js_urls, function() {\n",
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = fig()\n",
    "p.line(x_nnorm, y_nnorm)\n",
    "p.vbar(x_bin, top=vals, width=0.8)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Whilst this looks reasonable for $\\sigma$ not very far from its original value, the more it differs the worse the \"varied\" binomial distribution looks. There has to be another way.\n",
    "\n",
    "As our main goal was to decrease the standard deviation of a binomial distribution let us to begin with keep this method and make it available through a function call."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [],
   "source": [
    "def alberti_distribution(n, sigma):\n",
    "    \n",
    "    mu = n/2\n",
    "    # The new normal distribution\n",
    "    new_norm = normal_distribution(mu, sigma)\n",
    "    # The binomial distribution for n\n",
    "    bin = binomial_distribution(n)\n",
    "    # The normal distribution with the binomial distribution's sigma \n",
    "    bin_sigma = sqrt(sum(bin(i)*(i-mu)**2 for i in range(n+1)))\n",
    "    bin_norm = normal_distribution(mu, bin_sigma)\n",
    "    \n",
    "    def diff_to_integ(val, lower_bound=False):\n",
    "        def f(a):\n",
    "            # Set lower bound depending on whether it is specified or not\n",
    "            lower = val-a if not lower_bound else lower_bound\n",
    "            # Return value of integral minus the corresponding value of the binomial distribution\n",
    "            return integrate(norm, lower, val+a) - bin(val)\n",
    "\n",
    "        return f\n",
    "        \n",
    "    ## Calculating the interval boundaries\n",
    "    \n",
    "    interval_boundaries = []\n",
    "   \n",
    "    # Here we have to behave differently depending on whether n is even or odd\n",
    "    if int(mu) == mu:\n",
    "        a = solve(diff_to_integ(ceil(mu)), 0)[0]\n",
    "    else:\n",
    "        a = solve(diff_to_integ(ceil(mu), lower_bound=mu), 0)[0]\n",
    "        interval_boundaries.append(mu)\n",
    "    interval_boundaries.append(ceil(mu)+a)\n",
    "    \n",
    "    # Calculate the rest of the boundaries\n",
    "    for i in range(int(ceil(mu))+1, n):\n",
    "        lower_bound = interval_boundaries[-1]\n",
    "        a = solve(diff_to_integ(i, lower_bound=lower_bound), 0)[0]\n",
    "        interval_boundaries.append(i+a)\n",
    "    \n",
    "    # Mirroring for the other side\n",
    "    other_side = n - arr(interval_boundaries)[::-1]\n",
    "    # Another moment where a differentiation between even and odd n is to be made\n",
    "    if int(mu) == mu:\n",
    "        interval_boundaries = np.concatenate((other_side, interval_boundaries))\n",
    "    else:\n",
    "        interval_boundaries = np.concatenate((other_side, interval_boundaries[1::]))\n",
    "    # Adding +/- infinity to the list\n",
    "    interval_boundaries = list(interval_boundaries)\n",
    "    interval_boundaries.insert(0, -inf)\n",
    "    interval_boundaries.append(inf)\n",
    "       \n",
    "    # Calculating the corresponding integrals\n",
    "    vals = []\n",
    "    for i in range(n+1):\n",
    "        vals.append(integrate(new_norm, interval_boundaries[i], interval_boundaries[i+1])[0])\n",
    "    vals = arr(vals)\n",
    "\n",
    "    def f(x):\n",
    "        return vals[x]\n",
    "    \n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "only length-1 arrays can be converted to Python scalars",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-164-234aa41523b5>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mx_vals\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0marr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m21\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      6\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mvbar\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx_vals\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mtop\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdist3\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx_vals\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mwidth\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m0.8\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 7\u001b[1;33m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mline\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx_norm\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdist1\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mround\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx_norm\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcolor\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"red\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      8\u001b[0m \u001b[0mshow\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mp\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-162-8d1a4af0ba49>\u001b[0m in \u001b[0;36mf\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m     56\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     57\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 58\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mintegrate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnew_norm\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0minterval_boundaries\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0minterval_boundaries\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m+\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     59\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     60\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mTypeError\u001b[0m: only length-1 arrays can be converted to Python scalars"
     ]
    }
   ],
   "source": [
    "p = fig()\n",
    "dist1 = alberti_distribution(20, 10)\n",
    "dist2 = alberti_distribution(20, 5)\n",
    "dist3 = binomial_distribution(20)\n",
    "x_vals = arr(range(21))\n",
    "p.vbar(x_vals, top=dist3(x_vals), width=0.8)\n",
    "p.line(x_norm, dist1(round(x_norm)), color=\"red\")\n",
    "show(p)"
   ]
  }
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